Gibbs Sampling - Mathematical Background

Mathematical Background

Suppose that a sample is taken from a distribution depending on a parameter vector of length, with prior distribution . It may be that is very large and that numerical integration to find the marginal densities of the would be computationally expensive. Then an alternative method of calculating the marginal densities is to create a Markov chain on the space by repeating these two steps:

  1. Pick a random index
  2. Pick a new value for according to

These steps define a reversible Markov chain with the desired invariant distribution . This can be proved as follows. Define if for all and let denote the probability of a jump from to . Then, the transition probabilities are

p_{xy} = \begin{cases} \frac{1}{d}\frac{g(y)}{\sum_{z \in \Theta: z \sim_j x} g(z) } & x \sim_j y \\ 0 & \text{otherwise} \end{cases}

So

g(x) p_{xy} = \frac{1}{d}\frac{ g(x) g(y)}{\sum_{z \in \Theta: z \sim_j x} g(z) } = \frac{1}{d}\frac{ g(y) g(x)}{\sum_{z \in \Theta: z \sim_j y} g(z) } = g(y) p_{yx}

since is an equivalence relation. Thus the detailed balance equations are satisfied, implying the chain is reversible and it has invariant distribution .

In practice, the suffix is not chosen at random, and the chain cycles through the suffixes in order. In general this gives a non-stationary Markov process, but each individual step will still be reversible, and the overall process will still have the desired stationary distribution (as long as the chain can access all states under the fixed ordering).

Read more about this topic:  Gibbs Sampling

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